Conic Sections

Take a DOUBLE cone (two ice-cream cones tip to tip, extending forever) and cut it with a plane. The angle of the cut decides everything: a horizontal cut gives a circle; tilt the plane a little and the circle stretches into an ellipse.

Tilt the plane until it is exactly PARALLEL to the cone's slant side and the curve never closes up — that single borderline angle produces the parabola. It is the knife-edge between the closed curves (ellipses) and the doubly-open one (hyperbola).

Tilt past the parallel position (or cut straight down) and the plane slices BOTH halves of the double cone, producing the two branches of a hyperbola. The two branches are one curve — the same plane meeting the same cone, twice.

Standard form $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b$): the long semi-axis is $a$, the short one $b$, and the two foci sit at $(\pm c, 0)$ with $c^2 = a^2 - b^2$. The defining property: every point of the ellipse has distances to the two foci summing to $2a$ (a loop of string around two pins traces one).

$x^2 = 4py$: focus $(0, p)$, directrix $y = -p$, and every point of the curve is EQUIDISTANT from the focus and the directrix. Satellite dishes are parabolas because incoming parallel rays all bounce to the focus.

$\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ opens left-right with foci at $(\pm c, 0)$, $c^2 = a^2 + b^2$ (note the PLUS — foci sit outside), and asymptotes $y = \pm\tfrac{b}{a}x$: the branches hug these lines forever without touching. Defining property: the DIFFERENCE of the focal distances is constant ($2a$).